Lesson Velocity PVA, derivatives, anti-derivatives, initial value problems, object moving left/right and at rest

Position, Velocity, and Acceleration PVA problems

Using Anti-Derivatives with PVA

First find v(t) Now find s(3), v(3), | v(3) |, a(3) Next find s(t) ts(t)v(t)|v(t)|a(t)

ts(t)v(t)|v(t)|a(t)

An Object moving on a linear path (like a train on train tracks) The Object is not moving (at rest) when the Velocity is zero The Object is moving right when the velocity is positive The Object is moving left when the velocity is negative

Find v(t): Set v(t)=0, then solve for t to find where the particle is not moving: Interval test: v(t) + – Moves left (0, 2) and right (- , 0)  (2,  ) are Critical #’s ts –2 0 2 To Graph: t=2 t=0

Try: A particle moves along a linear path according the position equation s(t) = 4t 3 – 12t. Find when the particle is not moving, and the intervals when the particles moves left and right. V(t) = 12t 2 – 12 = 0 12(t + 1) (t – 1) = 0 Critical #’s t = 1, t = -1 particle is not moving Interval test: v(t) + – Moves left (-1, 1) and moves right (- ,-1)  (1,  )

Lesson Velocity Day 2: PVA with projectiles and gravity

Launched from ground level means the height s 0 = 0 Example: A projectile is launched vertically upward from ground level with an initial velocity of 80 ft/sec. A) Find the velocity at t = 1 sec and t = 3 sec. B) How high will the projectile rise? C) find the speed of the projectile when it hits the ground. Step 1: write the position: Step 2: write the velocity: Step 3: Find v(1) and v(3) v 0 = 80

Phrases to Look for

Example: An object is dropped from a bridge and hits the water below in 3 seconds. How many feet high was the object when dropped? write the position: What else do you know? Hits the water at t=10 seconds =>